Steady State Databehandling

Silica Rejection

2D Multiple Linear Regression

We assume that pH is the major variable that controls silica rejection. A liniear regression between the two shows correlation to some degree.

## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_SiO2 ~ pH, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.0437 -2.3231  0.2797  1.8346  4.7402 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -281.802     39.880  -7.066 5.88e-05 ***
## pH            32.493      4.155   7.820 2.65e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.25 on 9 degrees of freedom
## Multiple R-squared:  0.8717, Adjusted R-squared:  0.8574 
## F-statistic: 61.15 on 1 and 9 DF,  p-value: 2.654e-05
##         3 
## 0.4486406
## `geom_smooth()` using formula 'y ~ x'

## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_SiO2 ~ Cl_SO4, data = Ion_data_frame)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.0897  -5.9281   0.6586   6.9023  12.8352 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   25.856     15.261   1.694    0.124
## Cl_SO4         6.513     23.865   0.273    0.791
## 
## Residual standard error: 9.037 on 9 degrees of freedom
## Multiple R-squared:  0.008207,   Adjusted R-squared:  -0.102 
## F-statistic: 0.07447 on 1 and 9 DF,  p-value: 0.7911
## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_SiO2 ~ SiO2_mmol, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -11.168  -7.807   1.468   6.658  11.942 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   21.701     10.779   2.013   0.0749 .
## SiO2_mmol      5.092      6.447   0.790   0.4500  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.775 on 9 degrees of freedom
## Multiple R-squared:  0.06482,    Adjusted R-squared:  -0.03909 
## F-statistic: 0.6238 on 1 and 9 DF,  p-value: 0.45
## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_SiO2 ~ pH + SiO2_mmol, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.3524 -1.3066  0.3034  0.6636  2.5001 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -296.952     22.680 -13.093 1.10e-06 ***
## pH            33.045      2.340  14.119 6.15e-07 ***
## SiO2_mmol      6.082      1.345   4.522  0.00195 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.828 on 8 degrees of freedom
## Multiple R-squared:  0.9639, Adjusted R-squared:  0.9549 
## F-statistic: 106.9 on 2 and 8 DF,  p-value: 1.695e-06

## 
## Call:
## lm(formula = Rej_SiO2 ~ pH + Cl_anion, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.9471 -2.3996  0.3955  1.7761  4.6450 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -282.799     43.397  -6.517 0.000185 ***
## pH            32.532      4.421   7.358 7.93e-05 ***
## Cl_anion       1.070     10.538   0.102 0.921625    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.445 on 8 degrees of freedom
## Multiple R-squared:  0.8719, Adjusted R-squared:  0.8398 
## F-statistic: 27.22 on 2 and 8 DF,  p-value: 0.0002696
## 
## Call:
## lm(formula = Rej_SiO2 ~ pH + Cl_SO4 + SiO2_mmol, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8346 -1.3148 -0.1641  0.8955  2.8465 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -298.998     22.996 -13.002 3.71e-06 ***
## pH            32.953      2.364  13.939 2.31e-06 ***
## Cl_SO4         4.512      4.882   0.924  0.38610    
## SiO2_mmol      6.138      1.359   4.517  0.00274 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.845 on 7 degrees of freedom
## Multiple R-squared:  0.9678, Adjusted R-squared:  0.9541 
## F-statistic: 70.23 on 3 and 7 DF,  p-value: 1.371e-05

Conclusion: Silica rejection can be predicted from pH and concentration of silica on these samples.

3D Representation

Silica as function of pH and silica concentration

Silica as function of pH and Cl/SO4 ratio

Silica as function of Cl/SO4 ratio and silica concentration

Cl rejection

2D Multiple Linear Regression

We assume that pH does not play a significant role in Cl rejection, at least not to the same degree as for SiO2. Instead the chemical environment i.e. which other species are present and in what amounts are theorized to have a greater influence.

We check if [Cl],[anion],[Cl]/[anion] has a significant role Anion fraktion udregneds udfra Cl/[SO4]+[SiO2_ladet]+[Cl]

## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_Cl ~ Cl_SO4, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -10.864  -2.926   1.168   3.395   7.379 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   -6.613      9.657  -0.685    0.511
## Cl_SO4        25.183     15.103   1.667    0.130
## 
## Residual standard error: 5.719 on 9 degrees of freedom
## Multiple R-squared:  0.236,  Adjusted R-squared:  0.1511 
## F-statistic:  2.78 on 1 and 9 DF,  p-value: 0.1298
## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_Cl ~ Cl_anion, data = Ion_data_frame)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.1764  -3.1815   0.2664   3.4872   7.4276 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   -7.375     10.348  -0.713    0.494
## Cl_anion      28.524     17.523   1.628    0.138
## 
## Residual standard error: 5.751 on 9 degrees of freedom
## Multiple R-squared:  0.2274, Adjusted R-squared:  0.1416 
## F-statistic:  2.65 on 1 and 9 DF,  p-value: 0.138
## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_Cl ~ SO4_mmol, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -10.944  -2.997   1.103   3.457   6.812 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   15.228      3.944   3.861  0.00384 **
## SO4_mmol      -1.919      1.136  -1.689  0.12543   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.701 on 9 degrees of freedom
## Multiple R-squared:  0.2407, Adjusted R-squared:  0.1564 
## F-statistic: 2.854 on 1 and 9 DF,  p-value: 0.1254
## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_Cl ~ pH, data = Ion_data_frame)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -8.882 -3.099 -0.138  4.707  7.573 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -87.996     73.442  -1.198    0.261
## pH            10.134      7.652   1.324    0.218
## 
## Residual standard error: 5.986 on 9 degrees of freedom
## Multiple R-squared:  0.1631, Adjusted R-squared:  0.07009 
## F-statistic: 1.754 on 1 and 9 DF,  p-value: 0.2181
## `geom_smooth()` using formula 'y ~ x'

## 
## Call:
## lm(formula = Rej_Cl ~ SiO2_mmol, data = Ion_data_frame)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.6785 -4.9632  0.7619  3.2193 11.3016 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   18.933      7.312   2.589   0.0292 *
## SiO2_mmol     -5.986      4.373  -1.369   0.2043  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.953 on 9 degrees of freedom
## Multiple R-squared:  0.1723, Adjusted R-squared:  0.08029 
## F-statistic: 1.873 on 1 and 9 DF,  p-value: 0.2043
## 
## Call:
## lm(formula = Rej_Cl ~ SO4_mmol + total_sio2_ladet + Cl_mmol, 
##     data = Ion_data_frame)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -8.036 -2.382  1.022  2.983  7.982 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)  
## (Intercept)       -54.537     57.374  -0.951   0.3735  
## SO4_mmol           -2.680      1.330  -2.015   0.0837 .
## total_sio2_ladet   -3.580      8.933  -0.401   0.7005  
## Cl_mmol            15.422     12.920   1.194   0.2715  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.844 on 7 degrees of freedom
## Multiple R-squared:  0.3795, Adjusted R-squared:  0.1135 
## F-statistic: 1.427 on 3 and 7 DF,  p-value: 0.3135

## 
## Call:
## lm(formula = Rej_Cl ~ Cl_mmol * SO4_mmol * total_sio2_ladet, 
##     data = Ion_data_frame_both)
## 
## Residuals:
##        1        2        3        4        5        6        7        8 
##  2.28114  1.59695  0.52215 -2.88686 -2.26825 -4.42612  0.01679 -2.30095 
##        9       10       11 
##  4.56063  3.05730 -0.15277 
## 
## Coefficients:
##                                    Estimate Std. Error t value Pr(>|t|)
## (Intercept)                       -1044.434    941.557  -1.109    0.348
## Cl_mmol                             232.414    200.726   1.158    0.331
## SO4_mmol                            174.180    280.479   0.621    0.579
## total_sio2_ladet                    620.850   1059.632   0.586    0.599
## Cl_mmol:SO4_mmol                    -39.706     58.412  -0.680    0.545
## Cl_mmol:total_sio2_ladet           -144.214    223.126  -0.646    0.564
## SO4_mmol:total_sio2_ladet           -18.938    404.575  -0.047    0.966
## Cl_mmol:SO4_mmol:total_sio2_ladet     8.229     82.808   0.099    0.927
## 
## Residual standard error: 5.052 on 3 degrees of freedom
## Multiple R-squared:  0.8013, Adjusted R-squared:  0.3376 
## F-statistic: 1.728 on 7 and 3 DF,  p-value: 0.3526

## 
## Call:
## lm(formula = Rej_Cl ~ SO4_mmol + Cl_mmol + SiO2_mmol, data = Ion_data_frame_both)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.1347 -2.5512  0.2596  1.8279  5.7835 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -61.7651    37.9237  -1.629   0.1474  
## SO4_mmol     -2.6618     0.9648  -2.759   0.0281 *
## Cl_mmol      19.2005     8.4282   2.278   0.0568 .
## SiO2_mmol    -8.1470     3.4806  -2.341   0.0518 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.427 on 7 degrees of freedom
## Multiple R-squared:  0.6439, Adjusted R-squared:  0.4913 
## F-statistic:  4.22 on 3 and 7 DF,  p-value: 0.05328

Conclusion: [Cl] not useful to predict Rejection of Cl neither is [anion] or [SO4] [Cl]/[Anion] has a weak correlation to Rejection R2 0.53 (significant P>0.003)

[Cl/Anion]+pH R2 0.63 Cl/Anion P>0.00225 **,pH P>0.07329 . .

Removing the high SO4 point shows how wack the correlations otherwise are.

3D Representation

All Data Points

Cl rejection as function of Cl/SO4 ratio pH

## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'

Cl rejection as function of Cl/SO4 ratio and SiO2

## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'

Cl rejection as function of sio2 and pH

## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'

Removing of two outliers

Cl rejection as function of Cl/SO4 ratio pH

## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'

Cl rejection as function of Cl/SO4 ratio and SiO2

## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'

Cl rejection as function of sio2 and pH

## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'